Semantics and Reasoning Algorithms for a Faithful Integration of Description Logics and Rules

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Semantics and Reasoning Algorithms for a Faithful Integration of Description Logics and Rules. Boris Motik, University of Oxford. Contents. Why Combine DLs with LP? Main Challenge: OWA vs. CVA Existing Approaches Minimal Knowledge and Negation as Failure MKNF Knowledge Bases - PowerPoint PPT Presentation

Text of Semantics and Reasoning Algorithms for a Faithful Integration of Description Logics and Rules

  • Semantics and Reasoning Algorithmsfor a Faithful Integration ofDescription Logics and RulesBoris Motik, University of Oxford

  • ContentsWhy Combine DLs with LP?Main Challenge: OWA vs. CVAExisting ApproachesMinimal Knowledge and Negation as FailureMKNF Knowledge BasesReasoning and ComplexityConclusion

  • Description Logics and OWLOWL (Web Ontology Langage)language for ontology modeling in the Semantic Webstandard of the W3C (http://www.w3.org/2004/OWL/)OWL is based on Description Logics (DLs)inspired by semantic networksDLs have a precise semantics based on first-order logicswell-understood computational properties What can we say in DLs?

  • Missing Features (I)Relational expressivityOWL can express only tree-like axioms

    Polyadic predicatese.g., Flight(From, To, Airline)

    Can be addressed by rules (LP or ASP)9 S.(9 R.C u 9 R.D) v Q ,8 x:{[9 y: S(x,y) (9 x: R(y,x) C(x)) (9 x: R(y,x) D(x))] ! Q(x)},8 x,x1,x2,x3:{ S(x,x1) R(x1,x2) C(x2) R(x1,x3) D(x3) ! Q(x) }

  • Missing Features (II) Closed Worldsflight(MAN,STR)flight(MAN,LHR)flight(MAN,FRA)flight(FRA,ZAG)Question: is there a flight from MAN to MUC?Open worlds (=OWL):Dont know!

    We did not specify that we know information about all possible flights.Closed worlds (=LP):No.

    If we cannot prove something,it must be false.Partial solution: close off flight8 x,y: flight(x,y) $ (x MAN y STR) (x MAN y LHR) cannot express many things (e.g., transitive closure) Closed-world is orthogonal to closed-domain reasoningPerson v 9 father.Person Person(Peter) > v { Peter,Paul } CWA is available in various LP formalisms (e.g., ASP)

  • Missing Features (III) ConstraintsEach person must have an SSNnave attempt:Person u :(9 hasSSN.SSN) v ?in FOL, this is equivalent to:Person v 9 hasSSN.SSNassume that only Person(Peter) is givenwe expect the constraint to be violated (no SSN)but KB is satisfiable: Peter has some unknown SSNFOL formulaespeak about the general properties of worldscannot reason about their own knowledgeConstraints can be expressed in LP

  • Missing Features (IV)The heart is usually on the left, but in some cases it is on the rightNave approach:Human v HeartOnLeft Dextrocardiac v Human Dextrocardiac v :HeartOnLeft the class Dextrocardiac is unsatisfiablewith no contrary evidence, the heart is on the leftExceptionscannot be expressed in FOLcan be expressed in ASP

  • The Magic FormulaDLs (= taxonomical reasoning)+LP Rules (= relational expressivity + nonmonotonic inferences)=The Winning Combination!

  • ContentsWhy Combine DLs with LP?Main Challenge: OWA vs. CVAExisting ApproachesMinimal Knowledge and Negation as FailureMKNF Knowledge BasesReasoning and ComplexityConclusion

  • Open vs. Closed WorldsIn DLs we derive Person(a)The formula is equivalent to8 x : [Father(x) ! Person(x)]eliminates all models in which x is a father and not a personIn LP, : is interpreted as default negationread as is not provableThe example is unsatisfiableNegation defined using minimal knowledgeIt is illegal to state that someone is a father without stating that he is a person

    8 x : [Father(x) :Person(x) ! ?]Father(a)

  • Idea of Minimal KnowledgeDLsLPFather(a)M1 Father(a)M2Father(a), Person(a)M Father(a)8 x : [Father(x) :Person(x) ! ?]kills all models in which the formula does not holdAll models are of equal quality.This is the only minimal model.(There is no model M M.)We are left with models that contain Person(a)We are left with no model

  • Minimal Knowledge and NegationDLsRulesFather(a)8 x : [Father(x) :Person(x) ! Cat(x)]esures Cat(x) in each model where x is a father and not a personDoes not entail Cat(a)Does entail Cat(a), Cat(a)M1 Father(a) M2Father(a), Person(a)M Father(a), Cat(a)Nonmonotonic semantics typically prefer certain models.

  • ContentsWhy Combine DLs with LP?Main Challenge: OWA vs. CVAExisting ApproachesMinimal Knowledge and Negation as FailureMKNF Knowledge BasesReasoning and ComplexityConclusion

  • First-Order Rule FormalismsFirst-order combinations of DLs and rules:SWRL, CARIN, AL-log, DL-safe rulesA1 An B1 Bmconcepts (classes) = unary predicatesroles (properties) = binary predicatesinterpreted as first-order clausesSemantics is standard first-orderWoman(x) ! Person(x) and :Person(Lassie) imply :Woman(Lassie)Easily undecidabledecidability achieved by syntactic restrictions; e.g., DL-safetyIssues addressed:relational expressivity and polyadic predicatesnonmonotonic features

  • Loose Integrationdl-programs[Eiter, Ianni, Lukasiewicz, Schindlauer, Tompits, AIJ 2008]A B1 Bm not Bm+1 not BnA and Bi are first-order atoms over non-DL-predicatesBi can additionally be a query atom of the form DL[ S1 [ p1, S2 [ p2, S3 p3; Q ]Si DL predicatespi non-DL-predicatesQ a DL queryunderstand as conditional queries over a DL ontologyRules are layered over a DL KBthey do not contribute to DL consequences+--

  • Strong IntegrationA1 Ak B1 Bm not Bm+1 not BnDL+log [Rosati, KR2006]DL-atoms cannot occur under negation as failuresemantics:DL-predicates interpreted under OWAnon-DL-predicates interpreted under CWA no nonmonotonic reasoning over DL-predicatesdl-programs [Lukasiewicz, ESWC2007]no classical negation cannot capture ASPfaithful extension of LP and DLs only w.r.t. entailment of positive ground atomsunclear how to extend the semantics to make if faithful w.r.t. arbitrary consequences

  • Autoepistemic LogicsLP can be encoded into first-order AELAEL by [Konolige, Fund. Inf. 1991] Use AEL as a framework for integrating FOL and LP[de Bruijn, Eiter, Polleres, Tompits, IJCAI 2007]Various encodings proposed with different levels of faithfulnessconsiders disjunctive datalog and not ASPNo proof theory yet

  • ContentsWhy Combine DLs with LP?Main Challenge: OWA vs. CVAExisting ApproachesMinimal Knowledge and Negation as FailureMKNF Knowledge BasesReasoning and ComplexityConclusion

  • Knowledge Operator K(Researcher t Programmer)(Boris)Researcher v EmployedProgrammer v Employed Employed(Boris) Researcher(Boris) Programmer(Boris) K Employed(Boris) :K Researcher(Boris) :K Programmer(Boris)K is nonmonotonicif we assert Researcher(Boris), thenK Researcher(Boris) holds:K Researcher(Boris) does not hold any moreUsed in an algebra-like query language EQL-Lite[Calvanese, De Giacomo, Lembo, Lenzerini, Rosati, IJCAI 2007]

  • Default Negation Operator notBird(Tweety)K Bird(Tweety) not :Flies(Tweety) ! K Flies(Tweety)Interpreted as not consequenceRead as:ifTweety is a bird is a consequenceandTweety cannot fly is not a consequencethenTweety can fly should be a consequence

  • Minimal Knowledge and Negation as FailureSatisfiability defined w.r.t. an MKNF structure (I,M,N)I a FOL interpretationM and N sets of FOL interpretationsM is a model of if:(I,M,M) with I 2 M andfor each M M, there is some I 2 M such that (I,M,M) [Lifschitz, IJCAI 91; Artificial Intelligence 94]MKNF explains many nonmonotonic formalismsGelfond-Lifschitz reduct!

    (I,M,N) AiifA is true in I(I,M,N) :iif is false in I(I,M,N) 1 2iifboth 1 and 2 are true in I(I,M,N) K iif(J,M,N) for each J 2 M(I,M,N) not iif(J,M,N) for some J 2 N

  • ContentsWhy Combine DLs with LP?Main Challenge: OWA vs. CVAExisting ApproachesMinimal Knowledge and Negation as FailureMKNF Knowledge BasesReasoning and ComplexityConclusion

  • MKNF Knowledge BasesDL-safety:the rules are applicable only to explicitly named objectsH1 Hn B1, , BmHi are first-order or K-atomsBi are first-order, K-, or not-atomsP(t1, , tn)- first-order atomK P(t1, , tn)- K-atomnot P(t1, , tn)- not-atomMKNF Rule:MKNF Knowledge BaseK = (O, P)O a FOL KB in some language DLP a finite set of MKNF rulesSemantics by translation into MKNF(K) = K (O) r 2 P 8 x1,,xn : H1 Hn B1 Bm

  • Example (I)We derive seasideCity(Barcelona)assuming it does not lead to contradictionderiving seasideCity(Hamburg) would cause a contractionWe derive Suggest(Barcelona)this involves standard DL reasoningwe do not know the name of the beach in Barcelonadefault rule

  • Example (II)We treat in a special waywe minimize equality along with other predicatesthis yields intuitive consequencesThe constraint is satisfiedHolyFamily is a church,the architect of SagradaFamilia has been specified, andHolyFamily and SagradaFamilia are synonymsconstraint

  • FaithfulnessMKNF KBs are fully faithful w.r.t. DLs(O, ;) iff O for any FOL formula to achieve this, we modified MKNF slightlywe must treat equality in a special way

    MKNF KBs are fully faithful w.r.t. ASP(;, P) (:)A iff P (:)Afor A a ground atomalready shown by Lifschitz

    The combination seems quite intuitive

  • ContentsWhy Combine DLs with LP?Main Challenge: OWA vs. CVAExisting ApproachesMinimal Knowledge and Negation as FailureMKNF Knowledge BasesReasoning and ComplexityConclusion

  • How to Represent ModelsA MKNF model is a set of interpretations= typically infinite!we need a finite representationIdea: represent models by FOL formulaefind a first-order formula such thatM = { I | I }We represent using K-atoms(P,N) a partition of all K-atoms into positive and negativedefines the consequences that must hold in an MKNF modelobjective knowledge:obK,P = O [ { A | K A 2 P }our main task is to find a partition (P,N) that defines a model

  • Characterization of MKNF ModelsGroundingGuess a partition that defines an MKNF modelCheck whether the rules are satisfied in this model.Check whether this model is consistent with the DL KB.Check whether this is the model of minimal knowledge.Check whether the query does not hold in the model.These are the extensions to the standard algorithm for disjunctive

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